The Numerical Solution of Schrödinger's Equation

Abstract

Schr\"odinger's equation may be approximated to any desired accuracy by a difference equation over a lattice covering the region of integration. The solutions of this difference equation minimize a certain quadratic form (analogous to the energy integral $\ensuremath{\int}{\ensuremath{\psi}}^{*}H\ensuremath{\psi}$) subject to certain normalization and, for the higher states, orthogonality conditions. A practical numerical method is developed for the solution of this variation problem. By altering the values of a rough solution at each lattice point in turn by a simple improvement formula, the value of the quadratic form is continually decreased until the desired minimum is reached. Illustrations of the method are given for one-dimensional problems. Practical details are given for handling two-dimensional lattices, in particular for the solution of the problem of one electron in an axially symmetric field.